A square root should never be negative by convention or can be proved?

Question

I'm interested to know if there is any mathematical proof to show that square root should never be negative or it is just by convention ?

Answer 1

Well, as aforementioned, it is by definition that it cannot be negative. The following is a reason for why it is not considered negative (still dependent on a definition) in addition to the reason of your confusion:

Why Positive: First, extend the field from real numbers to complex numbers (namely, now, we consider a larger set of numbers represented by $r(cos \theta+i\cdot sin \theta)$ , r $\in \mathbb{R}$, $\theta \in [0,2\pi)$, $i=\sqrt{-1}$.) For simplicity, lets talk about square root of 1 (in the complex plane, any circle is geometrically similar to the unit circle, so the generality is conserved.) Then, square root is the second root of unity. Thus, if you consider $1$ as: $1(cos0+i \cdot sin0)$, the square root is $1 (cos(0/2)+i\cdot sin(0/2))$ which is still $1$; and if you consider $1$ as : $1(cos(2\pi)+ i\cdot sin(2\pi))$, the square root is $1(cos(2\pi/2)+i\cdot sin(2\pi/2))$ which is now $-1$. But take note that in the definition of (the polar form of) complex numbers, we excluded $\theta=2\pi$. Therefore, the second consideration, namely $\sqrt 1=-1$, by the definition of complex numbers is not valid.

Why confused: If we define square root as the inverse of the function $f(x)=x^2$, we would like to recover the fact that $f(-1)=f(+1)=1$. However, a function, in order to have an inverse needs to be one to one. And clearly, $f(x)=x^2$ is not one to one. But if you 'cut' the function at $x=0$ into two pieces, one for $x<0$ and the other for $x\geq0$, you'll acquire two one to one functions. Remembering that geometrical concept of finding the inverse of a function is reflecting the function's diagram with respect to the line $y=x$, the inverse of these two functions would appear as two branches of a horizontal parabola ($y^2=x$). Now, if we are talking about what values could yield to a specific x of this horizontal parabola, we should get a negative and a positive value. (of course except at zero, that these two values are equal.) But, if we are talking about a function, the horizontal parabola is obviously not a function... therefore, we need to choose one of those two branches mentioned above to be able to define a square root function. We chose the second half (the upper branch)... in the same way, we could choose to exclude $\theta=0$ instead of $\theta=2\pi$ in the "Why Positive" section. Then, it would mean that we need to consider the lower branch for consistency.

It may help you to check why this confusion doesn't happen when talking about the third root (and generally the odd roots)... [Hint: look at the diagram of $x^3$ for the inverse discussion and find the $0/3$, $1/3$, and $2/3$ angles for the complex number discussion.]

Answer 2

Let me quote from wikipedia:

For example, $4$ and $−4$ are square roots of $16$ because $4^2 = (−4)^2 = 16$. Every nonnegative real number a has a unique nonnegative square root, called the principal square root.

The principal square root is nonnegative by convention/ definition.

Answer 3

Indeed the term square root of actually means the positive root of. While both $4^2$ and $(-4)^2$ give 16, $\sqrt{16}=+4$, not $-4$.